decision making under time pressure: an independent test of ...decision making under time pressure...

Memory & Cognition1999, 27 (4), 713-725

Decision making is an inseparable component of all theactivities commonly studied by cognitive psychologists.This is manifested by the prominent role that decisionprocesses play in many areas of cognition, ranging fromperceptual processes (Link, 1992) to memory recogni-tion (Ratcliff, 1978) and categorization (Nosofsky & Pal-meri, 1997). Recently, a convergence of ideas has formedregarding the dynamic nature of the decision process thatunderlies all the above-mentioned cognitive abilities (cf.Ratcliff, Van Zandt, & McKoon, in press)—namely, theidea that activation for or against each action is sequen-tially sampled, from moment to moment, and accumulatedover time. The accumulation process continues until athreshold is reached, at which time a “winner takes all”action is triggered. This sequential sampling type of dy-namic decision process has been highly successful in ex-plaining the effects of time pressure on accuracy andspeed–accuracy tradeoff (see Link, 1992; Nosofsky & Pal-meri, 1997; Ratcliff et al., in press).

The broad success of sequential sampling models inperception, memory, and categorization has led some de-

cision researchers to explore their usefulness in the moretraditional domain of decision-making tasks, such as risk-taking decisions (Albert, Aschenbrenner, & Schmalhofer,1989; Aschenbrenner, Albert, & Schmalhofer, 1986; Buse-meyer, 1985; Diederich, 1995; Kornbrot, 1989; Petrusic& Jamieson, 1978; Wallsten, 1995). However, the majorityof decision researchers have not followed this lead, andmost of the studies in decision making have centered ontesting static models originating from economic ratherthan cognitive theory (see Goldstein & Weber, 1996). Con-sequently, there has been a neglect in conducting inde-pendent tests of the predictions made by sequential sam-pling models of risk-taking decisions. The purpose of thisarticle is to test some basic predictions regarding the ef-fects of time pressure on risk-taking decision making.

The risk-taking decision task used for this experimentwas developed by Dror, Katona, and Mungur (1998) andis a simplified variant of the game of blackjack. In thistask, the decision maker must decide whether or not togamble by taking another card from a deck in order to max-imize his or her total points without exceeding 21. Thistask was chosen because the information-processing de-mands (stimulus encoding and response production) areminimal and identical across trials, thus enabling us to iso-late the time pressure effects in the decision stage. Fur-thermore, this task allowed us to systematically manipu-late the levels of risk by varying the probability that takinga card would “bust” (exceed 21).

For this simple task, sequential sampling models makean a priori prediction regarding the effect of time pressure

713 Copyright 1999 Psychonomic Society, Inc.

This research was supported by grants funded by AFRL (awarded tothe first author) and by NSF Grant SBR-9602102 and NIMH Grant5R01MH55680 (awarded to the second author). We thank ZhengpingMa for help with some of the data analyses and Donna Stevens forproofreading the paper. The authors thank the reviewers for very help-ful comments. Correspondence concerning this article should be ad-dressed to I. E. Dror, Department of Psychology, Southampton Univer-sity, Highfield Southampton SO17 1BJ, England (e-mail: [email protected]). www: http://www.cogsci.soton.ac.uk/~dror

Decision making under time pressure:An independent test of sequential sampling models

ITIEL E. DRORSouthampton University, Highfield Southampton, England

JEROME R. BUSEMEYERIndiana University, Bloomington, Indiana

and

BETH BASOLASouthampton University, Highfield Southampton, England

Choice probability and choice response time data from a risk-taking decision-making task were com-pared with predictions made by a sequential sampling model. The behavioral data, consistent with themodel, showed that participants were less likely to take an action as risk levels increased, and thattime pressure did not have a uniform effect on choice probability. Under time pressure, participantswere more conservative at the lower risk levels but were more prone to take risks at the higher levelsof risk. This crossover interaction reflected a reduction of the threshold within a single decision strat-egy rather than a switching of decision strategies. Response time data, as predicted by the model,showed that participants took more time to make decisions at the moderate risk levels and that timepressure reduced response time across all risk levels, but particularly at the those risk levels that tooklonger time with no pressure. Finally, response time data were used to rule out the hypothesis that timepressure effects could be explained by a fast-guess strategy.

714 DROR, BUSEMEYER, AND BASOLA

on the frequency of choosing the gamble (details are pre-sented in the Appendix): Time pressure will increase thefrequency of choosing a gamble when the risk (proba-bility of losing) is high but will decrease the frequencyof choosing the gamble when the risk (probability of los-ing) is low. In other words, sequential sampling modelspredict a crossover interaction effect between time pres-sure and risk level on frequency of choosing the gamble.Thus, time pressure is predicted not to have a uniform ef-fect (of being either more conservative or more prone totake risks) but, rather, a polarization in terms of behavior:At the low risk, people turn more conservative and takefewer gambles, whereas at the high risk, they are morerisky and take more gambles.

The sequential sampling model also makes specificpredictions for this task concerning the mean responsetimes for two different types of responses. A congruentresponse occurs if a card is chosen when there is very lowrisk (the response is compatible with the risk level); anincongruent response occurs if a card is not chosen whenthere is very low risk (the response is not compatible withthe risk level). The sequential sampling model predictsthat incongruent responses take a longer time to makethan congruent responses (see the Appendix for details).

A second reason for selecting this task is that a num-ber of studies have investigated the blackjack game, iden-tifying optimal versus heuristic strategies (Anderson &Brown, 1984; Bond, 1974; Keren & Wagenaar, 1985; Phil-lips & Amrhein, 1989). The optimal strategy for maxi-mizing the probability of winning is to take an additionalcard if the current total falls below an optimal cutoff cri-terion, where this criterion is determined in a complexway that takes into account the card shown for the op-posing player. A common heuristic strategy for this taskis the never-bust strategy, which is a conservative tendencyto avoid taking an additional card and to stay with thecurrent total. Although nonoptimal, the never-bust strat-egy is attractive because it avoids the possibility of ac-tively taking an action that may be directly responsiblefor losing the game.

Changes from optimal toward simpler heuristic strate-gies under time pressure provide an alternative explana-tion for the effects of time pressure on risk taking (see,e.g., Johnson, Payne, & Bettman, 1995; Payne, Bettman,& Luce, 1996). However, the predictions inferred fromstrategy switching differ from the predictions derived fromsequential sampling models for this particular task. It ispredicted that, in switching to a never-bust strategy undertime pressure, the frequency of choosing the gamble neverincreases (because the likelihood of switching to the sim-pler never-bust strategy increases under time pressure).

Another plausible hypothesis is that participants switchto a fast-guess strategy under time pressure. That is, anindividual will occasionally make fast random guesseswhen under time pressure. Like the sequential samplingmodel, this fast-guess hypothesis predicts a crossover in-

teraction for choice probability. But unlike the sequentialsampling model, the fast-guess model predicts faster re-sponses for incongruent than for congruent responses(see the Appendix for details).

The study reported here includes two parts. The firstpart reports the main empirical results obtained from col-lecting behavioral data on decision making under timepressure. To achieve this goal, we tested participants onDror et al.’s (1998) risk-taking decision-making task undera time pressure condition and under a no time pressurecondition. This was done independently of the secondpart, which consisted of generating predictions for bothchoice probability and decision time through the use ofa sequential sampling model (Busemeyer & Townsend,1993) and comparing the predictions to the actual be-havioral data. In the Discussion section, we review relatedresearch on decision making under time pressure andcontrast the predictions of sequential sampling modelswith alternative explanations based on changes in heuris-tic strategies under time pressure.

METHOD

ParticipantsThirty-two participants (16 males, 16 females) took part in the

study for credit in an undergraduate psychology course.

MaterialsThe decision task was a computer-simulated card game similar to

the game of blackjack, where the goal is to maximize the sum ofcards without going over 21. The modified blackjack task was de-signed for several purposes. First, it was a simplification of the fa-miliar blackjack game (e.g., in our task, we excluded aces to avoidthe ambiguity of whether they should be counted as 1s or 11s andexcluded jacks, queens, and kings to avoid issues relating to percep-tual recognition of the face cards; we also did not allow “splitting”and other options that are part of the blackjack game). Second, theprobabilities associated with each choice are easily manipulated bythe values of the cards dealt to the players, and decisions are easilyquantified by the time it takes to make a binary decision (to take anadditional card or not).

On each trial, the participant received two cards, which appearedat the top half of the computer screen, and the computer received asingle card, which appeared at the bottom half of the screen. Eachcard was 3.2 � 4.9 cm, with its number displayed a single time inthe center of the frame, using a 36-size font, Geneva typeface.

The player’s cards were manipulated to vary the level of risk pro-duced by the probability that the player’s total would exceed 21 (andlose the hand). At the very low end of the risk level spectrum werethe trials with a sum of 11 or less, in which case there was no riskin taking an additional card (regardless of the value of the additionalcard, participants could not go over 21). There were also trials withlow risk (trials with sums of 12 or 13), trials with medium risk (tri-als with sums of 14 or 15), trials with high risk (trials with sums of16 or 17), trials with very high risk (trials with sums of 18 or 19),and trials with infinite risk (trials with a sum of 20, in which case itis wrong to take an additional card, because participants would nec-essarily go over 21 and lose their entire hand).

The value showing on the computer’s card was also manipulatedon each trial. The computer’s card was programmed to have a lowlevel (card values of 2, 3, or 4), a moderate level (card values of 5,

DECISION MAKING UNDER TIME PRESSURE 715

6, or 7), or a high level (card values of 8, 9, or 10). A high level forthe computer card was used to increase the total sum that the playerneeded to beat the computer and, consequently, increase the play-er’s need to take another card. A low level was used to decrease thetotal sum that the player needed to beat the computer and, conse-quently, reduce the player’s need to take another card.

Trials were counterbalanced to ensure an equal number of trialswith all the possible combinations of player’s and computer’s cards.For each possible sum of the participants’ two cards (17 possible sums,from 4 to 20), a group of 9 trials was constructed, giving a total of153 trials. Each of the 9 trials in every group was matched to one ofthe nine possible values of the computer’s card (2–10). The possiblecombinations of the participants’ cards were also counterbalanced(e.g., for the total sum of 14, all the following combinations wereused as equally as possible: 4 and 10, 5 and 9, 6 and 8, 7 and 7, 8and 6, 9 and 5, and 10 and 4). For administering the task, the 153trials were organized in a sequence of nine blocks, each consistingof 17 trials. Each block contained a single presentation of all the pos-sible sums of cards, and the order of the trials within the blocks wasrandomized. (Although this counterbalancing procedure has exper-imental design advantages, it has the disadvantage of influencing thefrequencies in such a way that each triple of cards is not equally likely.)

ProcedureThe participants were tested individually in a single testing ses-

sion. Half the participants were first tested with no time pressureand then with time pressure; the other half of the participants weretested in the reverse order (in each group of participants who weretested in a certain order—either time pressure first or no time pres-sure first—half the participants were male and half were female).For both the time pressure and the no time pressure conditions, theparticipants were tested on the identical 153 experimental trials, ex-cept that the order of the trials was different (trials were randomizedwithin each block). Time pressure was created by asking the par-ticipants to respond as quickly as possible. It was emphasized thattheir response time was critical. Under the no time pressure condi-tions, the participants were told to take as much time as they neededto respond and were asked to carefully consider their decisions. Theparticipants were seated approximately 45 cm from the computerscreen. Instructions appeared on the computer screen, followed bythree practice trials.

At the onset of each trial, an exclamation point was displayed onthe computer screen. When the participants were ready to begin thetrial, they pressed the space bar. Three cards appeared in each trial;two were considered to be dealt to the participant, and one to the op-ponent. On the basis of the value of these cards, the participant de-cided whether or not to take an additional card (“splitting,” “insur-ance,” and other options from the game of blackjack were notallowed). If a new card was taken, the value of the new card wasadded to the participant’s total; if the participant decided not to takeanother card, the total remained unchanged. The participants’ goalwas to have the sum of their cards be higher than their opponent’s,without exceeding 21.

The participants were instructed to assume that, after they madetheir decision, they and their opponent would have the opportunityto take additional cards. They were also informed that they wouldnot see what additional card they received, what additional cardsthe opponent received, or who won the hand (this was done to en-sure that participants would not change decision strategy or crite-rion during the task as a result of recent positive or negative out-comes of previous decisions; see Dror, Rafaely, & Busemeyer, 1999,and Rafaely, Dror, & Busemeyer, 1998, for details). Rather, aftermaking a decision, the participants would be presented on the nexttrial with a new and independent sample of cards. If the participantschose to take an additional card, they pressed the yes key (the “b”key, which was labeled yes). If the participants did not want an ad-ditional card, they pressed the no key (the “n” key, which was la-

beled no). After the participants responded, a blank screen appearedfor 350 msec, followed by an exclamation point to signal the begin-ning of a new trial. The participants responded yes and no by usingtwo fingers of their dominant hand and pressed the space bar withtheir nondominant hand. The participants did not receive any feed-back or additional information. Throughout the instructions andpractice trials, the participants were encouraged to ask questions,and clarifications were given by the experimenter as needed. How-ever, no talking was allowed during the actual experiment.

RESULTS AND DISCUSSION

Empirical DataFirst, choice probabilities (proportion of trials in

which an additional card was requested) are reported,using the player’s risk level (1 � no risk, 2 � low risk,3 � medium risk, 4 � high risk, 5 � very high risk, 6 �infinite risk), the computer’s card level (1 � low level,2 � moderate level, 3 � high level), and time pressure (nopressure vs. pressure) as within-subjects factors. Second,response times (pooled across choices) are described,using the same three factors. Finally, conditional responsetimes (computed separately for gambling and not gam-bling responses) are shown, using risk category (low �risk levels 1, 2, or 3 vs. high � risk levels 4, 5, or 6), timepressure (pressure vs. no pressure), and response cate-gory (choose to gamble vs. choose not to gamble) as fac-tors. (Computer card was omitted because of sample sizelimitations.)

Choice probability. As is illustrated in Figure 1 (leftpanels), choice probability decreases as risk level in-creases. More important, the curve within each panel forthe no time pressure condition is steeper than the corre-sponding curve for the time pressure condition, produc-ing a crossover interaction between risk level and timepressure. This crossover result confirms the a priori pre-diction made by sequential sampling models.

A specific test of the critical risk level � time pres-sure crossover interaction was performed by computinga single degree of freedom contrast, using the logistictransformed choice proportions as follows. We computedthe difference between the no time pressure and timepressure conditions at two different levels of risk, levels2 and 5. These two levels were chosen because they aremoderately extreme, but not to the extent that the choiceprobabilities approach zero or one. At level 2, the differ-ence was predicted by the sequential sampling models tobe positive, and at level 5 it was expected to be negative.We observed the following results: First, the differencebetween the no pressure and the pressure conditions forthe risk level 2 produced a positive contrast equal to+.56; second, the difference between the no pressure andthe pressure conditions at risk level 5 produced a nega-tive contrast equal to �.32; finally, the contrast of thesetwo differences was significant, according to a t test[t(155) � 10.08, p < .0001].

Another notable finding is that the computer’s cardlevel moderated the effects of risk level and time pres-sure: When there was no time pressure, a low level for


Figure 1. The left three panels show the probability of taking an additional card by the participants as a functionof risk level. The top, middle, and bottom panels are for low, moderate, and high computer card levels. In all panels,there is a backward-S-shaped curve as risk level increases, and the time pressure curve is flatter than the no time pres-sure curve. Furthermore, there is a crossover interaction between the curves for the time pressure and no time pres-sure conditions in each of the panels. These effects are moderated by the computer card level, but only for the no timepressure condition. The right three panels show the predictions of the model for choice probability under the sameexperimental conditions. The steeper curve represents the predictions when the inhibitory threshold is set to a highcriterion and the bound is increased, and the flatter curve represents the predictions when the inhibitory thresholdis set to a low criterion (under time pressure) and the bound is smaller. The precise quantitative form of the curvesdepends on specific parameter values estimated from the data, but the crossover interaction pattern and the effectof the computer card level is a parameter-free prediction of the model.


the computer’s card suppressed the tendency to gambleat the low risk levels, whereas a high level for the com-puter’s card enhanced the tendency to gamble at the lowrisk levels; however, when there was time pressure, thecomputer’s card had little or no effect on the choice prob-ability. If the data for the time pressure condition is re-plotted with choice probability as a function of risk leveland a different curve for each computer card level, thethree curves lie virtually right on top of each other. Thisreflects the fact that the computer card level had no ef-fect under the time pressure condition and suggests thatthe participants did not have sufficient time and resourcesto consider the computer’s card.

The trends in Figure 1 are supported by statistical testsobtained from a three-way repeated measures analysis ofvariance (ANOVA) using the logistic transformed choiceproportion for each subject and condition as the depen-dent variable. The main effect of risk level was signifi-cant [F(5,155) � 178.10, MSe � 0.514, p < .0001], therisk level � time pressure interaction was significant[F(5,155) � 4.68, MSe � 0.061, p < .0005], and the risklevel � time pressure � computer card level interactionalso was significant [F(10,310) � 2.32, MSe � 0.053,p � .0121]. The main effect of computer card level wassignificant [F(2,62) � 57.31, MSe � 0.097, p < .0001],the computer card level � risk level interaction was sig-nificant [F(10,310) � 15.29, MSe � 0.076, p < .0001], andthe computer card level � time pressure interaction alsowas significant [F(2,62) � 3.29, MSe � 0.055, p � .0440].

Response time. As is illustrated in Figure 2 (left pan-els), the mean response time is an inverted-U-shapedfunction of risk level. However, the level of this curve ismuch lower and the slope is much flatter under the timepressure condition than under the no time pressure con-dition. Finally, the peaks of the curve shift from left to rightas the computer card level increases from low to high.

The trends in Figure 2 are supported by statistical testsobtained from a three-way repeated measures ANOVA,using the mean response time for each subject and con-dition as the dependent variable. The main effect of risklevel was significant [F(5,155) � 20.92, MSe � 1,010,309,p < .0001], the risk level � time pressure interaction wassignificant [F(5,155) � 11.5, MSe � 642,320, p < .0001],and the risk level � time pressure � computer card levelinteraction also was significant [F(10,310) � 1.86, MSe �243,049, p � .0498]. The main effect of computer cardlevel was significant [F(2,62) � 3.23, MSe � 294,584,p � .0465], and the computer card level � risk levelinteraction was significant [F(10,310) � 2.90, MSe �274,504, p � .0017].

Conditional response time. Figure 3 (top panel) showsthe mean response times plotted as a function of risk cat-egory, with a separate line for each response categoryand time pressure condition. As can be seen in the fig-ure, there is a crossover interaction between risk categoryand response category, but only for the no time pressurecondition. The crossover interaction under the no timepressure condition can be summarized as follows: The

decision to gamble takes less time than the decision notto gamble when the risk category is low, but the decisionto gamble takes more time than the decision not to gam-ble when the risk category is high. Recall from Figure 1that choice probability follows the opposite pattern: Thedecision to gamble is more frequent than the decision notto gamble under the low risk category, but the decision togamble is less frequent than the decision not to gambleunder the high risk category. This replicates previous find-ings that have shown an inverse relation between the prob-ability that an alternative is chosen and the time requiredto choose it (Busemeyer, 1982; Petrusic & Jamieson, 1978).

The trends in Figure 3 are supported by statistical testsobtained from a three-way repeated measures ANOVA,using the conditional mean response time for each sub-ject and condition as the dependent variable. The riskcategory � time pressure � response category interactioneffect was significant [F(1,17) � 5.557, MSe � 271,476,p < .05]. (Note: the denominator degrees of freedom re-flect missing observations owing to the fact that somesubjects never chose one of the alternatives under someconditions. The analysis with missing cells was com-puted with an SAS system type III sums of squares; seeSAS manual, 1996.)

Sequential Sampling Model PredictionsThe specific sequential sampling decision model used

to generate predictions was derived from decision fieldtheory (Busemeyer & Townsend, 1993; Townsend &Busemeyer, 1996), which is the only sequential samplingmodel that has been mathematically formalized specifi-cally for risky decision-making tasks. Hence, it enabled usto generate quantitative as well as qualitative predictionsfor the results of this experiment. Furthermore, decisionfield theory has been successful in explaining a wide rangeof fundamental findings from research on risk-taking de-cision making (see Busemeyer & Townsend, 1993).

Figure 4 illustrates the basic process that is assumedto occur within a single trial of this decision task. Thetrial begins at time t � 0 with the presentation of the play-er’s two cards and the computer’s card; then the playerbegins the process of deciding whether or not to take an-other card on this trial. The process begins with an ini-tial preference state, denoted P(0), which is equal to zero,assuming that the player starts out unbiased (i.e., P(0) �0). Then, from moment to moment, the player imaginesand anticipates the positive or negative outcomes of eachchoice, given the cards that are presented. At one mo-ment, the player may anticipate winning by taking an ad-ditional card, strengthening the approach tendency, sothat P(t � 1) � P(t) > 0. At a later moment, the decisionmaker may imagine losing the hand by taking an addi-tional card and going over 21, strengthening the avoidancetendency so that P(t � 2) � P(t � 1) < 0.

This vacillation in tendencies continues until eitherthe approach tendency becomes strong enough to exceedan inhibitory threshold to trigger the action of requestingan additional card [P(t) > �θ] or the avoidance tendency


Figure 2. The left three panels show the mean response time (pooled across gambling and not gambling responses)that participants needed to decide whether or not to take an additional card as a function of risk level. The top, mid-dle, and bottom panels are for low, moderate, and high computer card levels. In all panels, the participants neededmore time at the difficult trials (with medium levels of risk) and less time at the easy trials (when the risk level wasrelatively high or low). With time pressure, the inverted-U-shaped response time curve was flatter and lower than thecurve with time pressure. The locations of the curves’ peaks are influenced by the computer card level. The right threepanels show the predictions of the model for mean response time (pooled across responses) for the same experimen-tal conditions. The model accurately predicts the inverted-U-shaped relations between mean choice time and risk levelfor each time pressure condition.


becomes strong enough to exceed an inhibitory thresh-old to trigger the action of declining an additional card[P(t) < �θ]. The final choice on a particular trial is deter-mined by whether the approach tendency or the avoid-ance tendency crosses the threshold first. The decisiontime is determined by the time it takes to exceed thethreshold magnitude.

According to the model of the decision process de-scribed above, choice and decision time are controlledby two factors: One is θ, the magnitude of the inhibitorythreshold; the second is d, the mean change in preferencestate at each moment. The magnitude of the inhibitorythreshold, θ, is assumed to be determined by the timepressure manipulation: The threshold is reduced underthe high time pressure condition. The inhibitory thresh-old controls the amount of time spent thinking about the

decision, which in turn controls the amount of informa-tion accumulated about the anticipated outcomes pro-duced by each action. Increasing the threshold increasesthe average decision time. Thus, more information is ac-cumulated, which causes greater decision accuracy (theprobability of choosing the optimal response that maxi-mizes the likelihood of winning). Decreasing the thresh-old (under time pressure) reduces the average decisiontime. Thus, less information is accumulated, which causesdecreased decision accuracy. (See Busemeyer, 1985, fora more formal analysis, or the Appendix.)

The sign of the mean change in preference state, d, de-termines whether the approach or avoidance tendencygrows stronger on the average over time. If d is positive,the approach tendency (take another card) grows strongerover time than the avoidance tendency, and if d is nega-

Figure 3. The top panel shows the mean response time participants needed foreach response alternative (to take a card and not to take a card—the conditionalresponse time) as a function of risk level. There is a crossover interaction betweenthe two, but only for the no time pressure condition. The bottom panel shows thatthe model accurately predicts the behavioral data.


tive, the avoidance tendency (decline another card) growsstronger over time than the approach tendency.

Recall that the probability of exceeding 21 and losingthe game by taking another card increases as the player’stotal increases, but the need to take another card to beatthe computer increases as the value of the computer’scard increases. These two factors are represented in themodel by assuming that the mean change in preferencestate, d, is a decreasing function of the risk level and anincreasing function of the computer card level. This im-plies that, for each fixed value of the computer card level,the probability of choosing to gamble will be a decreas-ing function of the risk level (see the Appendix for details).However, under the time pressure condition, there maybe insufficient time to consider the computer’s card, so inthis case, the computer’s card is ignored. (Changes in at-tention over time are consistent with Diederich’s [1995]multiattribute generalization of decision field theory.)

The mathematics used to derive the formulas for thisparticular model have been presented elsewhere (Buse-meyer & Townsend, 1992, 1993), and the Appendix pro-vides the derived formulas used to compute the predic-tions shown below. It is important to note that strongquantitative tests of the model are made possible throughthe use of both choice probability and choice responsetime measures of preference. The basic method for test-ing the model is first to estimate the model parametersfrom the choice probability data and then to use thesesame parameters to generate parameter-free predictionsfor the mean response time data.

More specifically, seven model parameters were esti-mated from the 36 observed choice proportions, and theseparameter estimates were used in the formulas to com-pute the choice probabilities from the model. The criti-cal test is obtained by entering these same seven param-eters into the formulas for computing the predictions forchoice response time data. This provides a generaliza-tion test of the model by producing predictions for all ofthe response time data that do not require estimating anynew parameters.

First, the right-hand panels of Figure 1 illustrate thefits to the choice probability data. As can be seen by com-paring the left and the right panels, the model correctlyreproduces the crossover interaction between risk leveland time pressure, and it also reproduces the effects ofcomputer card level on choice probability. Although thespecific form of the curves producing the crossover inter-action depends on the estimated parameters, the backward-S-shaped form of the function, as well as the crossoverinteraction, is a necessary property of the model. In par-ticular, if the data failed to show this crossover, themodel would be unable to fit the results.1

Second, the right-hand panels of Figure 2 illustrate thepredictions for the response time data. Note that thesepredictions are made without using any parameters (thisincludes the time unit for the predictions, so only the pat-tern of the predictions is important). As can be seen bycomparing the left and the right panels, the model cor-rectly predicts the inverted-U-shaped function relating re-sponse time and risk level, and the model also correctly

Figure 4. A model of the task based on decision field theory. The horizontal axis rep-resents the time interval, starting with the initial presentation of the two cards, untila final decision is reached on whether or not to take an additional card. The verticalaxis represents the participant’s relative preference, where a positive preference is atendency to take another card and a negative preference is a tendency to avoid takinganother card. The jagged curve in the figure represents the evolution of the partici-pant’s preference state over time, P(t). The action of taking an additional card or notis determined by which threshold is exceeded on that trial, and the decision time is de-termined by the time interval required to exceed that threshold (the vertical line attime T ). As the bounds decrease (when the thresholds are lowered under time pres-sure), the time interval to exceed a threshold is smaller.


predicts the proportional reduction in response time ateach risk level produced by the time pressure manipula-tion. Finally, the model correctly predicts the shift in thepeak of this U-shaped function with increases in thecomputer card level.

A strong test of the model is obtained by evaluatingthe predictions for decision times conditioned on thespecific choice (the response chosen). The model cor-rectly predicts that the mean time to take another card isshorter than the mean time to choose to not take another

card, when the risk level is relatively low. The model alsocorrectly predicts the opposite order when the risk levelis relatively high. Furthermore, the model predicts thatthis crossover interaction effect is attenuated by timepressure (see the bottom part in Figure 3).

Strategy-Switching Model PredictionsAn alternative explanation for the effects of time pres-

sure on decision behavior is that subjects switch to sim-pler heuristic strategies under time pressure. Two possi-

Figure 5. The unfilled bars represent the mean response time for a congruent re-sponse, and the dark filled bars represent the mean response time for the incongru-ent response. The first pair represents the no time pressure condition, and the secondpair represents the time pressure condition. The top panel represents the observeddata, the middle panel represents the predictions of the original sequential samplingmodel, and the bottom panel represents the predictions of the fast-guess model.


bilities were discussed earlier, the never-bust strategyand the fast-guess strategy. The crossover interaction ef-fect observed in Figure 1 clearly rules out the never-buststrategy, but it does not rule out the fast-guess strategy.A critical test of the fast-guess hypothesis can be obtainedby examining the predictions for congruent and incon-gruent mean response times. In general, the fast-guessmodel predicts fast incongruent responses, whereas theoriginal sequential sampling model predicts slow incon-gruent responses (see the Appendix for details).

More specifically, the fast-guess model asserts that,under time pressure, participants will make a fast randomguess on some proportion of trials. To test this model, newparameters were estimated from the sequential samplingmodel by fitting it to the choice data and mean responsetime data with one major change (see the Appendix fordetails). Instead of decreasing the threshold bound undertime pressure conditions, this parameter was held con-stant, and a new parameter was added, representing theprobability of fast guessing under time pressure. In sum,seven parameters (six original plus one fast guess) wereestimated from the choice probability data, and thesesame parameters were used to make predictions regard-ing decision time. This model produced essentially thesame fit to the choice probability data as the original se-quential sampling model.

The critical test of the sequential sampling model ver-sus the fast-guess model is shown in Figure 5. The toppanel shows the observed mean time to make congruent(white bar) versus incongruent (dark bar) choices, wherea congruent choice was to take another card under verylow risk (levels 1 or 2) and an incongruent choice wasthe decision not to take a card under these same condi-tions.2 The left pair of bars represents the no time pressurecondition, and the right pair represents the time pressurecondition. Note that the incongruent responses were mademore slowly than the congruent responses under the crit-ical time pressure condition.3

The middle panel shows the corresponding predictionsof the original sequential sampling model. As can beseen in the figure, the original sequential sampling modelaccurately predicts the pattern of observed results underboth time pressure conditions. The bottom panel showsthe corresponding predictions produced by the fast-guessmodel. As can be seen, the fast-guess model fails to de-scribe the pattern of results under the time pressure condi-tion. In sum, the finding of slower incongruent responses,as compared with congruent responses, is contrary to thefast-guess model and supports the original sequentialsampling model.

GENERAL DISCUSSION

Recent emphasis has been placed on the decision pro-cess in numerous successful models from a variety ofcognitive domains (Link, 1992; Nosofsky & Palmeri,1997; Ratcliff, 1978; Smith, 1995). It seemed interest-ing to examine this decision process in the more “pure”

decision-making domain. This is especially importantinasmuch as the above-mentioned models all have simi-lar conceptualizations of what constitutes the decision-making process—namely, sequential sampling over timeuntil a certain threshold is met.

We collected behavioral data on a risk-taking decision-making task (Dror et al., 1998) to independently test thesetypes of models. Because sequential sampling modelshave considered the effects of time pressure (Link, 1992;Nosofsky & Palmeri, 1997; Ratcliff, 1978), we includeda time pressure component in our behavioral task. Weused decision field theory (Busemeyer & Townsend, 1993;Townsend & Busemeyer, 1996) to generate predictions,because it is a sequential sampling model that has beenmathematically developed for risky decision makingand, thus, it enabled us to make precise predictions forour behavioral gambling-like task.

Our behavioral data were consistent with previous be-havioral research (Dror et al., 1998), showing that par-ticipants systematically were less likely to take a card asthe risks increased and that response time for making thedecisions decreased as the risks were toward the ends ofthe risk level spectrum. Our manipulation of time pres-sure caused a polarization effect; participants were moreconservative and less likely to take an action (request anadditional card) at the lower levels of risk and were morerisky and likely to take an action at the higher levels ofrisk. That is, the S-shaped choice probability curve, as afunction of risk, was flatter under the time pressure con-dition. Response times decreased with time pressure,and the inverted-U-shaped choice time curve, as a func-tion of risk, was also flatter under the time pressure con-dition. Finally, the time to make a choice was inverselyrelated to the probability of making that choice, repli-cating previous findings (Busemeyer, 1982; Petrusic &Jamieson, 1978).

The empirical results described above were consistentwith the predictions generated by decision field theory.The theoretical analyses based on this model indicatedthat time pressure produced two different effects on thedecision process. One was to reduce the threshold boundfor the cumulative preference strength needed to make adecision, and the second was to reduce the amount of at-tention given to less important dimensions in the task.The effect of time pressure on the threshold bounds isconsistent with a long-standing assumption of decisionfield theory (Busemeyer, 1985). The effect of time pres-sure on attention to less important dimensions is consis-tent with recent generalizations of decision field theoryto multidimensional choice problems (Diederich, 1995).

An alternative hypothesis, based on the idea of strat-egy switching, was also evaluated. Many recent studies(for an overview, see Svenson & Maule, 1995) have at-tempted to explain changes in choice behavior under timepressure by changes in decision strategies, from a moreaccurate and time-consuming strategy to a less accuratebut simpler and faster heuristic strategy (see, e.g., John-son et al., 1995; Payne et al., 1996). Keren and Wagenaar


(1985) have identified two such strategies in blackjack-type tasks (see, also, Anderson & Brown, 1984; Bond,1974; Phillips & Amrhein, 1989). First, the basic strat-egy, which is the optimal strategy that maximizes theprobability of winning, requires the player to considerhis or her own cards as well as the computer’s card. Sec-ond, the never-bust strategy is a conservative tendency toavoid taking an additional card and to stay with the cur-rent total. Note that the never-bust strategy only requiresattention to the player’s cards and does not require atten-tion to the computer’s card. Although nonoptimal, thenever-bust strategy is attractive because it is simple and,more important, avoids the possibility of actively takingan action (requesting an additional card) that may be di-rectly responsible for losing the game.

The basic or optimal strategy for our risk-taking taskis to always choose to take an additional card if the cur-rent total sum of the cards falls below a cutoff criterion(this criterion depends on, among other things, the com-puter’s card). In other words, if we were to plot the choiceprobability curve predicted by this strategy (similar to thatshown in Figure 1), it would be a backward step functionwith an abrupt jump from one to zero at the cutoff. How-ever, if this cutoff varies across participants, the curve pro-duced by averaging across participants would appear asa backward-S-shaped curve like that shown in Figure 1.

The never-bust strategy is a very simple heuristic strat-egy of not taking an additional card if there is some riskthat the player’s new total will exceed 21. It is reasonableto expect that, under time pressure, at least on some pro-portion of trials, participants may abandon the more dif-ficult process of locating the optimal cutoff for the sim-pler heuristic strategy that guarantees not losing the entirehand by taking an additional card. However, this strategy-switching model does make one important distinguish-ing prediction. That is, the curve produced under timepressure must lay completely below that predicted by notime pressure (because the proportion of trials in whichparticipants use the simpler strategy of not taking an ad-ditional card always increases under time pressure). How-ever, as is apparent in Figure 1 (top panel), our data donot support this prediction.

Another plausible heuristic strategy is to resort to fastrandom guesses under time pressure. This fast-guessmodel explains the time pressure � risk level crossoverinteraction effect on choice probability as being the re-sult of mixing random guesses on some proportion of tri-als, which flattens out the backward-S-shaped curve.However, the fast-guess model incorrectly predicts thatsubjects will make faster incongruent, as opposed tocongruent, responses under time pressure, which is con-trary to the observed results.

In sum, for this particular task, the never-bust and fast-guess strategies are the most common and obvious heuris-tics, and we can rule out the hypothesis that participantstend to switch to these particular strategies under time

pressure. In contrast, the sequential sampling model pro-vides a comprehensive explanation of both the choiceprobability and the choice response time behavioral data.

Of course, it is impossible to enumerate all conceivablestrategies, so we cannot completely eliminate the generalform of this explanation. Moreover, one could interpretchanges in attention to the opponent’s card as a kind ofstrategy switch. But then the notion of strategy becomesso broad that one has to question its testability or pre-dictive utility.

One final comment is that our results do not rule outother possible effects of time pressure on decision pro-cesses, such as a speed-up of each processing step (seeBen Zur & Breznitz, 1981). But note that simply speed-ing up processing by a constant factor under time pressurecannot explain the changes in the pattern of choice prob-abilities and response times produced by time pressure.

It is interesting to note that our empirical results, ob-tained from a traditional decision-making task, appear tobe very similar to the speed–accuracy tradeoff resultsfound in other cognitive tasks, such as perception (seeLink, 1992), memory (see Ratcliff, 1978), and catego-rization (see Nosofsky & Palmeri, 1997). Furthermore,the success of sequential sampling models in accountingfor the results obtained from this risk-taking decision-making task strengthens our confidence in the general useof this decision process across a variety of cognitive tasks(for further discussion on the usefulness of such models,see Dror & Gallogly, 1999). In summary, both the em-pirical results and the theoretical analyses presented inthis study support the view that common principles of de-cision processes underlie a wide range of cognitive tasks.

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NOTES

1. Note that the above model assumes that time pressure has two sep-arate effects on the decision process. One is a decrease in the thresholdbound, and the second is a decrease in attention to the opponent’s card.One might question whether both of these assumptions are really nec-essary. To answer this question, the model was refit under two differentconstraints. On the one hand, when the model was refit assuming a de-crease in threshold but no decrease in attention under time pressure, the

model failed to produce the observed time pressure � opponent’s cardinteraction effect. On the other hand, when the model was refit assum-ing a decrease in attention but no decrease in threshold under time pres-sure, the model failed to produce the observed crossover interaction be-tween time pressure and risk level. In sum, both assumptions wererequired to explain the choice probability results.

2. The two extreme low risk level conditions were used for this testfor two reasons. The first reason is that the two most extreme levels pro-vide the most diagnostic and the clearest definitions for congruent andincongruent responses. Second, the experimental design provided 90trials per subject under the two lowest levels, but it provided only 18 tri-als per subject under the two highest levels. The reader should also notethat the data shown in Figures 3 and 5 overlap but are not the same. Forexample, in Figure 3, the open square plotted under the low risk cate-gory represents mean time to choose no card, averaged over risk levels1, 2, and 3; in Figure 5, the black bar above the time pressure label rep-resents the mean time to choose no card, averaged over risk levels 1 and2. Risk level 3 was included in Figure 3 in order to include all of the datain the basic analyses. Risk level 3 was not included in Figure 5 becausethis risk level was too high to treat choosing no card as an incongruentresponse. In other words, risk level 3 does entail some risk, which makesit an ambiguous case for defining the choice of no card as an incongru-ent response.

3. A dependent t test was used to perform a statistical test of the hy-pothesis implied by the fast-guess model. Define µC and µI as the pop-ulation mean response times for the congruent and incongruent re-sponses, respectively, under time pressure. To falsify the fast-guess model,we need to reject the directional hypothesis H0: (µI � µC ) < 0. Thesample mean difference between congruent and incongruent responsetimes under time pressure produced a t statistic that exceeded the con-ventional cutoff for rejecting this hypothesis [t(31) � 1.8, p < .05].

APPENDIX

The purpose of this appendix is to present the formulas usedto compute the predictions from the model derived from deci-sion field theory for this task. Decision field theory has sevenstages, but it was sufficient to use only stage three (Equations3c and 3d in Busemeyer & Townsend, 1993) for this simpletask. In this case, the formula for the probability of choosing togamble for an individual, denoted Pr[G], derived from the gen-eral theory for this task, is

Pr[G] � 1/[1 � exp(�2θd )]. (1)

Note that this is an S-shaped logistic function of the mean changein preference state, d, and that the threshold, θ, moderates theslope of this logistic function. Recall that d is inversely relatedto the risk level, and therefore, choice probability is predictedto be a backward-S-shaped function of risk level. Increasing thetime pressure decreases θ, which decreases the slope of the lo-gistic function of d, and this produces the predicted crossoverinteraction.

The mean time to make a choice for an individual subject,denoted E[T ], derived from the general theory for this task, is

E[T ] � (θ /d ) (2Pr[G] � 1). (2)

Conceptually, the ratio (θ /d ) can be interpreted as the well-known formula, travel time equals distance divided by rate oftravel. Under time pressure, the distance to travel θ decreases,so that the time to travel also decreases. Note that this equationdoes not include a time unit for the decision process, and so thetime constants for the predicted and observed times are notequated. Thus, the predicted times are only proportional to theobserved times.

Fitting the choice probability data entailed estimating sevenparameters (detailed below), by searching for values that min-


imized the sum of squared deviations between the predicted andthe observed choice probabilities. Two of the seven parameterswere used to estimate two threshold values (θ � 1.44 for the notime pressure condition, and θ � 0.955 for the time pressurecondition).

Four parameters were used to determine the mean change inpreference, d, for each of the 36 conditions, as follows. Denotedi jk as the mean change in preference under risk level i(i � 1,2, 3, 4, 5, 6), computer card level j ( j � 1, 2, 3), and time pres-sure condition k (k � 0,1). The following expression was usedto determine the mean change:

dijk � [(1 � k) cj � a(i � 3.5)]. (3)

When k � 0 (no time pressure), the mean change equals the dif-ference between the effect of the computer card level and the ef-fect of the risk level; when k � 1 (time pressure present), the ef-fect of the computer card is eliminated, and the mean change isinversely related to the effect of the risk level. The effect of risklevel was assumed to be proportional to the risk level, with theconstant of proportionality estimated from the choice data (a �.842). Three parameters were estimated from the choice data torepresent the effect of the computer card level (c1 � 2.85, c2 �2.92, c3 � 3.14).

An additional seventh parameter was used to incorporate in-dividual differences into the model. This was accomplished byadding a random effect Sl to Equation 3:

dijkl � [(1 � k) cj � a (i � 3.5 � Sl)]. (4)

Conceptually, negative values of Sl represent individuals thatare more risk seeking than the average person and positive val-ues of Sl represent individuals who are more risk aversive thanthe average person. In general, the distribution of individualdifference effects is unknown. For simplicity, the individual dif-ference effects Sl were represented by a binomial distributionacross the values {�1.5, �1.0, �0.5, 0, 0.5, 1.0, 1.5} with aparameter p � .42 (estimated from the choice data). Specifi-cally, the probability of Sl was computed by

P[Sl � s] � [(6!)/(2s � 3)!(3 � 2s)!] p2s � 3(1 � p)3 � 2s. (5)

To summarize, the choice probabilities and mean responsetimes were computed separately for each condition and eachvalue of the individual difference effect Sl, using Equations 1,2, and 4, and then these predicted values were averaged acrossthe individual differences Sl for each condition, using the prob-abilities defined by Equation 5. The seven parameters were es-timated with the choice probability data alone, and then thesesame seven parameters were used to compute the mean re-sponse times shown in Figure 3 and Figure 5.

The fast-guess model is obtained by employing Equations 1and 2 directly for the no time pressure conditions and by modify-ing Equations 1 and 2 by mixing a guessing probability, g, undertime pressure:

Pr[G] � (1 � g) /[1 � exp(�2θd )] � (.5)g, (6)

and

E [T ] � (1 � g) (θ/d ) (2Pr[G] � 1) � g � Tg , (7)

where Tg is the mean time required to make a fast guess. The meantime to make a fast guess cannot be estimated from the choicedata alone. Therefore, it was necessary to estimate eight pa-rameters from both the choice and the mean response time data:a threshold, θ, constant across time pressure conditions; a guess-ing rate g under time pressure; three coefficients for the com-puter card; two coefficients, a and p, described earlier, used todetermine the mean change for each condition and individual;and, finally, the mean time to make a guess, Tg . These eight pa-rameters were then used to compute the mean times to makecongruent and incongruent responses, shown in Figure 5.

As Figure 5 shows, the predictions computed from the deci-sion field model indicate that the mean response time undervery low risk is slower for incongruent responses than for con-gruent responses for both time pressure conditions. The con-ceptual reason for this prediction is that, when the risk level isvery low, incongruent responses tend to be chosen when the meanchange in preference for an individual (Equation 4) is small inmagnitude; when this occurs, the rate of travel (see Equation 2)slows down the time reach the boundary for both time pressureconditions. For example, when the risk level is very low but anindividual is very risk aversive, the mean change in preferencewill be close to zero; this is the case that is most likely to pro-duce an incongruent response, but this response also tends totake more time because of the slow rate of travel.

In contrast, the predictions computed from the fast-guessmodel indicate that incongruent responses are faster than con-gruent responses under time pressure and very low risk. Theconceptual reason for this prediction is that the fast-guess modelassumes that a change in decision strategy occurs under timepressure. Thus, it predicts the same result as the decision fieldmodel with no time pressure. But under time pressure, the in-congruent responses tend to be produced by the fast guesses,which makes the incongruent responses faster than the congru-ent responses (the latter tend to occur after waiting to reach thecriterion bound).

(Manuscript received March 31, 1998;revision accepted for publication July 6, 1998.)

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